Subgroups — #22582677

Order 8991.

460 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15688532	C₃	1	16	yes	maximal, complemented
2	#15688531	C₃	1	8	yes	maximal, complemented
3	#15688530	C₃	1	2	yes	maximal, complemented
4	C₃×#11758606	C₃	1	4	yes	maximal, complemented
5	3₋¹⁺²×C₃₇⋊C₃	C₃	1	8	yes	maximal, complemented
6	H(3,𝔽₃)×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
7	C₃₇×#737	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₃₇⋊C₃	C₃²	1	4	yes	—
9	3₋¹⁺²×C₃₇	C₃²	1	8	yes	complemented
10	H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
11	C₃×C₉×C₃₇	C₃²	1	4	yes	complemented
12	C₃²⋊(C₃₇⋊C₃)	C₃²	1	8	yes	—
13	#11758608	C₃²	1	16	yes	complemented
14	C₃×#66524	C₃²	1	8	yes	complemented
15	#11758606	C₃²	1	8	yes	—
16	#11758606	C₃²	1	1	yes	characteristic
17	#11758606	C₃²	1	8	yes	complemented
18	#11758606	C₃²	1	16	yes	complemented
19	#11758605	C₃²	1	24	yes	complemented
20	C₉×C₃₇⋊C₃	C₃²	1	24	yes	complemented
21	C₃²×C₃₇	C₃³	1	4	yes	—
22	C₃×C₃₇⋊C₃	—	9	4	no	—
23	C₃×C₃₇⋊C₃	C₃³	1	8	yes	—
24	C₃×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
25	C₉×C₃₇	C₃³	1	8	yes	—
26	C₉×C₃₇	C₃³	1	1	yes	characteristic
27	#66524	C₃³	1	16	yes	—
28	#66524	C₃³	1	2	yes	—
29	3₋¹⁺⁴	—	37	1	no	maximal, Sylow
30	C₃×C₃₇	—	3	4	no	—
31	C₃×C₃₇	C₃⁴	1	1	yes	characteristic
32	C₃₇⋊C₃	—	3	1	no	—
33	C₃₇⋊C₃	—	3	8	no	—
34	#737	—	37	16	no	—
35	#737	—	37	2	no	—
36	#737	—	37	8	no	—
37	#737	C₃₇⋊C₃	1	1	yes	characteristic, complemented
38	C₃×3₋¹⁺²	—	37	8	no	—
39	C₃×3₋¹⁺²	—	37	4	no	—
40	C₃×H(3,𝔽₃)	—	37	1	no	—
41	C₃₇	3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
42	C₃³	—	37	4	no	—
43	3₋¹⁺²	—	37	16	no	—
44	3₋¹⁺²	—	37	24	no	—
45	3₋¹⁺²	—	37	8	no	—
46	3₋¹⁺²	—	37	1	no	—
47	3₋¹⁺²	—	37	16	no	—
48	3₋¹⁺²	—	37	8	no	—
49	3₋¹⁺²	C₃×C₃₇⋊C₃	1	8	yes	complemented
50	H(3,𝔽₃)	—	37	8	no	—
51	H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
52	C₃×C₉	—	37	8	no	—
53	C₃×C₉	—	37	24	no	—
54	C₃×C₉	C₃×C₃₇⋊C₃	1	4	yes	complemented
55	C₃²	—	37	8	no	—
56	C₃²	—	37	1	no	—
57	C₃²	—	333	4	no	—
58	C₃²	C₃²×C₃₇⋊C₃	1	4	yes	—
59	C₉	—	37	2	no	—
60	C₉	—	37	16	no	—
61	C₉	C₃²×C₃₇⋊C₃	1	8	yes	—
62	C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
63	C₃	—	3	4	no	—
64	C₃	—	111	1	no	—
65	C₃	—	111	8	no	—
66	C₃	C₃³×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃	C₃³×C₃₇⋊C₃
commutator, socle	C₃×C₃₇	C₃⁴
Fitting	C₃₇×#737	C₃
hypercenter	#737	C₃₇⋊C₃
hypocenter	C₃₇	3₋¹⁺⁴
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ #737

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #15688532	C₃	semidirect
2. #15688531	C₃	semidirect
3. #15688530	C₃	semidirect
4. C₃×#11758606	C₃	semidirect
5. 3₋¹⁺²×C₃₇⋊C₃	C₃	semidirect
7. C₃₇×#737	C₃	semidirect
9. 3₋¹⁺²×C₃₇	C₃²	semidirect
11. C₃×C₉×C₃₇	C₃²	semidirect
13. #11758608	C₃²	semidirect
14. C₃×#66524	C₃²	semidirect
17. #11758606	C₃²	semidirect
18. #11758606	C₃²	semidirect
19. #11758605	C₃²	semidirect
20. C₉×C₃₇⋊C₃	C₃²	semidirect
37. #737	C₃₇⋊C₃	semidirect
41. C₃₇	3₋¹⁺⁴	semidirect
49. 3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
54. C₃×C₉	C₃×C₃₇⋊C₃	semidirect